On standard \(L\)-functions attached to alt\(^{n-1}(\mathbb{C}^ n)\)-valued Siegel modular forms (Q1911578)

It is shown that the completed Dirichlet series \[ \Lambda(s,f,st) = \Gamma_\mathbb{R} (s+1) \left\{\prod^{n-1}_{j=1} \Gamma_\mathbb{C} (s+k+1-j) \right\} \Gamma_\mathbb{C} (s+ k-n) L(s,f,st) \] has a meromorphic continuation to the whole \(s\)-plane provided \(k\) is even, \(n\) odd and \(2k\geq n>2\). Moreover it satisfies the usual functional equation \[ \Lambda (s,f,st) = \Lambda (1-s,f,st) \] and is an entire function for \(k>n\). Here \(f\) is an \(\text{alt}^{n-1} (\mathbb{C}^n)\)-valued cusp form for the Siegel modular group \(Sp(n, \mathbb{Z})\) with respect to the irreducible rational representation \(\text{det}^k \otimes \text{alt}^{n-1}\) of signature \((k+1,k+1, \dots, k+1,k)\) and an eigenfunction of the Hecke algebra. Clearly \(L\) denotes the standard \(L\)-function associated to \(f\) and \(\Gamma_\mathbb{R}\) resp. \(\Gamma_\mathbb{C}\) suitably modified \(\Gamma\)-functions. As the author remarks this result was previously known except for the representations \(\text{det}^k\) and \(\text{det}^k \otimes \text{sym}^\ell\).